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Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index \(H\geq\frac 1 4\). (English) Zbl 1059.60067
The paper is devoted to generalized covariation processes and an Itô formula related to the fractional Brownian motion. The paper follows “almost pathwise calculus techniques” developed by Russo and Vallois, and it reaches the \(H=\frac{1}{4}\) barrier, developing very detailed Gaussian calculations. One motivation of this paper is to prove an Itô-Stratonovich formula for the fractional Brownian motion with \(H\geq\frac{1}{4}\). Such a process has, in some sense, a finite 4-variation and a finite pathwise \(p\)-variation for \(p>4\). It was even proved that the cubic variation is, in some sense, zero, when the Hurst index is bigger than \(\frac{1}{6}\). The main achievement is the proof of the existence of the 4-covariation \([g(B^H),B^H,B^H,B^H]\) for \(H\geq\frac{1}{4}\), \(g\) being locally bounded. Moreover, it is proved that this covariation is Hölder continuous with parameter strictly smaller than \(\frac{1}{4}\). The result provides, as an applications, the Itô-Stratonovich formula for \(f(B^H)\), \(f\) being of class \(C^4\) and a generalized Bouleau-Yor formula for fractional Brownian motion. Some results for local time are also obtained. The technique used here is a “pedestrian” but accurate exploitation of the Gaussian feature of fractional Brownian motion.

60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
60G15 Gaussian processes
60G48 Generalizations of martingales
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